What Nyquist Didn't Say

For audio signal processing, the filters are generally active, so no inductors are used. [speaker crossovers excepted.]

In many applications, ringing cannot be tolerated. Scales for instance.

ISTR Roger Lagadec at Studer (and Sony) pointing this out in the early

1980's, after listening tests on early digital systems with piano music showed that something was amiss. Probably in AES and IEEE archives.

martin

Ring before the main part of the signal arrives. FIR filter responses are often shown as if zero delay were the middle of the filter, effectively subtracting out the constant delay added by the filter.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

... snip ...

Don't think of a single signal. An impulse (or a step) has a wide range of frequency components. Some are delayed more than others when passing through the filter.

--
 Some informative links:

Please read more carefully. The filter rings before the main part of the output step *emerges* but after the step arrives at the input. The filter's inherent delay makes that quite possible.

Jerry

-- "The rights of the best of men are secured only as the rights of the vilest and most abhorrent are protected." - Chief Justice Charles Evans Hughes, 1927 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Its only non-causal if it arrives before it was sent. :-)

There is no reason why the main signal should not arrive latter than some of the crud which accompanies it. Its perfectly normal in a dispersive medium.

Steve

May I ask what software you used to render the maths on that page? It looks clearer than the stuff I produce. MathML is getting into browsers now, but the rendering of that looks so bad with anything I have tried, that inserted images in HTML pages still seems the only practical approach.

I'd still like to see a web page I can point people to when they say a

10kHz sine wave on a CD will come out as a square wave/triangular wave/some other weird notion.

Steve

=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF Here is the line verbatim: "Linear phase has a very undesirable side effect, it rings *before and after* the step, supposed to be more audible."

Nothing wrong with my reading. Now if you are somehow looking at the output to interpret where the large transition occurred, that is a different story. However, any filter where the impulse response goes negative will have such ringing, be it linear phase or not. You need to visualize the convolution.

SciLab

Well, get writing!

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

It's called *pre-ringing* and it appears because the chunks are processed forward and backward in a row, so a unity pulse will have an identical rising and falling edge. If the filter is of the ringing type, thus the ringing occurrs twice. You are right in saying it's impossible, but only in an analog world. Digital filters do have a latency which will always be longer than the delay of the corresponding analog filter; with linear phase it will be twice the FIR size plus twice the conversion time and more than double than the analog counterpart. Well done analog filters are of the *minimum phase* type, having just the lowest possible delay for that shape of output response. This is possible to realize digitally with IIR filters only. And do not think that even a Gauss filter has only positive FIR-coefficients. This would be only true for a filter of infinite length, which apparently isn't that desirable at all. For practicable sizes the location of the poles and zeros has to be modified and one might get even negative coefficients, depending on the ratio of sampling- and filter frequency and filter length.

--
ciao Ban
Apricale, Italy

=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF

lay

log

to

The Gaussian to which I refer is S domain. If you mapped it to Z domain, it would have to be IIR, not FIR.

Even steep linear phase analogue filters will exhibit pre-ringing. If you were to linearise the phase response of, say, a Butterworth filter, by adding one or more all-pass sections, its impulse repsonse will ring before and after the main output. Of course, the overall delay must go up for the filter to remain causal.

Jeroen Belleman

Joerg wrote: (someone wrote)

(and be distinguished on the other end).

The important point being that the math is the same even though the goal is different. I suppose, then, the sample rate should be a lemma to Nyquist's telegraph channel theorem.

By the way, Gauss published the first paper on the FFT.

-- glen

...

Minimum-phase (or nearly minimum) FIRs are possible, just not symmetric FIRs. You can make maximum-phase FIRs too. Then *all* the ringing is on the leading edge.

Jerry

--
        "The rights of the best of men are secured only as the
        rights of the vilest and most abhorrent are protected."
            - Chief Justice Charles Evans Hughes, 1927
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(Actually, Gauss never published it. It was only published posthumously as part of his notes.)

I like it.

As for section 1, for a periodic signal, or one that you only care about over a finite time, you can (mathematically) sample perfectly in a finite time. Realistically, quantum mechanics and the uncertainty principle, in other words noise, will get to you.

The question of < or

Then, Gauss wrote the first published paper on FFT?

If you want to put it that way, very few people publish papers, they just send them to someone else to publish.

But yes, I had forgotten that.

-- glen

Hello Glen,

We have to remember what means there were back in their days. Far fewer journals with available space. No word processors. Very costly type-setting process. Etc.

Even nowadays publishing isn't easy. I have done a few and the whole process is quite laborious. However, we now have an excellent means of publishing just about anything (legal) we want: The web. Everybody can set up a web site and go ahead. Also, you can publish your ideas in newsgroups just like this one. All that provides instant publication. Gauss, Nyquist and others didn't have all this and I assume Shannon was too far into retirement by then as well. AFAIR he passed away at old age around five years ago.

--
Regards, Joerg

http://www.analogconsultants.com

No. Gauss' work was not published until 1866, as a part of his collected works. Prior to that, there were various authors who published related algorithms (e.g. a paper by Everett in 1860, one published by Archibald Smith in 1846, and one published by F. Carlini in 1828, although these works only described restricted cases).

What does seem to be true is that Gauss was the first *recorded* discoverer of an FFT. He was also (apparently) the only author until Cooley & Tukey in 1965 to describe a general mixed-radix algorithm for any composite size.

(See the excellent paper, "Gauss and the History of the Fast Fourier Transform," by Heideman et al., IEEE ASSP Magazine, p. 14, October

1984.)

You're being a bit too pedantic for my taste; by "publish" in science, we usually mean "initiate the publication process".

Regards, Steven G. Johnson